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THE NUMBER OF ATOMIC MODELS OF UNCOUNTABLE THEORIES
Published online by Cambridge University Press: 08 February 2018
Abstract
We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with 
$ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that there is a complete theory in a language of size 
${\aleph _1}$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with 
$ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that for every complete theory T in a language of size 
${\aleph _1}$, if T has uncountable atomic models but no constructible models, then T has 
${2^{{\aleph _1}}}$ atomic models of size 
${\aleph _1}$.
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References
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